Cross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem

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dc.creatorAndersson, J.
dc.creatorWeiss, G. S.
dc.date2005-08-10
dc.date.accessioned2026-07-07T06:24:43Z
dc.date.available2026-07-07T06:24:43Z
dc.descriptionWe investigate singular and degenerate behavior of solutions of the unstable free boundary problem $$Δu = -χ_{\{u>0\}} .$$ First, we construct a solution that is not of class $C^{1,1}$ and whose free boundary consists of four arcs meeting in a {\em cross}-shaped singularity. This solution is completely unstable/repulsive from above and below which would make it hard to get by the usual methods, and even numerics is non-trivial. We also show existence of a degenerate solution. This answers two of the open questions in a recent paper by R. Monneau-G.S. Weiss.
dc.identifierhttps://arxiv.org/abs/math/0508184
dc.identifierhttp://arxiv.org/abs/math/0508184
dc.identifierdoi:10.1016/j.jde.2005.11.008
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/96640
dc.subjectAnalysis of PDEs
dc.subject35R35; 35J60
dc.titleCross-shaped and Degenerate Singularities in an Unstable Elliptic Free Boundary Problem
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